Harmonic Analysis Associated with the Jacobi–Dunkl Operator on $$(-\pi ,\pi )$$: Exotic Cases

Abstract

AbstractWe investigate some spectral properties of a second order differential-difference operator $$J_{\alpha ,\beta }$$ J α , β on $$L^2((-\pi ,\pi ),d\mu _{\alpha , \beta })$$ L 2 ( ( - π , π ) , d μ α , β ) , $$\alpha ,\beta \in \mathbb {R}$$ α , β ∈ R , called the Jacobi–Dunkl operator of compact type. Using an idea of Hajmirzaahmad, in exotic cases, e.g. when at least one of the two parameters $$\alpha ,\beta $$ α , β is $$\le -1$$ ≤ - 1 , we construct exotic orthonormal bases that consist of eigenfunctions of $$J_{\alpha ,\beta }$$ J α , β . This allows one to consider natural self-adjoint exotic extensions of $$J_{\alpha ,\beta }$$ J α , β and the corresponding exotic Jacobi–Dunkl and Jacobi–Dunkl–Poisson semigroups.